Bending-torsion vibration model with two-end energy dissipation

摘要

This research is devoted to the asymptotic and spectral analysis of a coupled Euler–Bernoulli and Timoshenko beam model. The model is governed by a system of two coupled differential equations and a two parameter family of boundary conditions modelling the action of self-straining actuators. The aforementioned equations of motion together with a four-parameter family of boundary conditions form a coupled linear hyperbolic system, which is equivalent to a single operator evolution equation in the energy space. That equation defines a semigroup of bounded operators. The dynamics generator of the semigroup is our main object of interest. For each set of the boundary parameters, the dynamics generator has a compact inverse. If all boundary parameters are not purely imaginary numbers, then the dynamics generator is a non-self-adjoint operator in the energy space. We calculate the spectral asymptotics of the dynamics generator. We find that the spectrum lies in a strip parallel to the horizontal axis, and is asymptotically close to the horizontal axis––thus the system is stable, but is not uniformly stable. We have also found that the set of the eigenfunctions of the dynamics generator forms an unconditional basis (the Riesz basis) in the state space of the system. The Riesz basis property allows us to solve explicitly boundary and/or distributed controllability problems via the spectral decomposition method.